-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathSBachPlant.m
444 lines (340 loc) · 19.1 KB
/
SBachPlant.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
classdef SBachPlant < DrakeSystem
% Defines the dynamics for the SBach rc plane
% Jan 5, 2015.
properties
end
methods
function obj = SBachPlant()
obj = obj@DrakeSystem(12,0,4,12,0,1);
obj = obj.setOutputFrame(obj.getStateFrame);
end
function xdot = dynamics(obj,t,x,u)
% % min/max protection here
% u(1) = min(850, max(150, u(1)));
% u(2) = min(850, max(155, u(2)));
% u(3) = min(850, max(180, u(3)));
% u(4) = min(860, max(160, u(4)));
% @param t time
% @param x state: x =
% Plane's X coordinate faces forward, Y to the right, and Z down.
% x(1):x (Forward Position, Earth frame)
% x(2):y (East or y position, Earth frame)
% x(3):z (z position (down), Earth frame)
% x(4):phi (roll angle)
% x(5):theta (pitch angle)
% x(6):psi (yaw angle)
% x(7):U (X-velocity, body frame)
% x(8):V (Y-velocity, body frame)
% x(9):W (Z-velocity, body frame)
% x(10):P (Angular velocity in X direction, body frame)
% x(11):Q (Angular velocity in Y direction, body frame)
% x(12):R (Angular velocity in Z direction, body frame)
%
% u(1):thr (Throttle command)
% u(2):ail (Aileron Command)
% u(3):elev (Elevator Command)
% u(4):rud (Rudder command)
%% Parameters fit from data %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Rotational inertias
Jx_fac = 1.7837;
Jy_fac = 4.2679;
Jz_fac = 6.9248;
% Throttle/propwash
thr_to_sp_ail = 2.3940;
thr_vel_fac_ail = 1;
thr_to_sp_elev = 1.2296;
thr_vel_fac_elev = 1;
thr_fac = 1;
% Elevator
elev_lift_fac = 1;
% Stabilizer
stab_force_fac = 0.1;
% Body drag
body_x_drag_fac = 0;
body_y_drag_fac = 0;
body_z_drag_fac = 0;
% Rate dependent force
F_Q_fac_x = 0; %
F_Q_fac_z = 0; %
% Throttle aerodynamic drag
thr_drag_fac = 0; % thr_drag_fac*1e-6;
% Rate dependent moments
M_P_fac = 0;
M_Q_fac = 0;
M_R_fac = 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Rotational inertias
Jx = Jx_fac*.0005; % The numbers are just guesses to get things in the right ball park
Jy = Jy_fac*.0009;
Jz = Jz_fac*.0004;
J = diag([Jx,Jy,Jz]);
invJ = diag([1/Jx,1/Jy,1/Jz]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Measured parameters (MKS units):
wing_area_out = 7848/(1000*1000);% m^2
wing_area_in = (2.7885*10^3)/(1000*1000); % m^2
wing_total_area = 2*wing_area_out + 2*wing_area_in;
out_dist = 132.8/1000; % Moment arm of outer wing section
in_dist = 46.4/1000; % Moment arm of inner wing section
ail_out_dist_x = 41/1000; % These are behind the COM
ail_in_dist_x = 65/1000; % These are behind the COM
elev_area = 5406/(1000*1000); % m^2
elev_arm = 240/1000;
rudder_area = (3.80152*1000)/(1000*1000);
rudder_arm = 258.36/1000; % m
stab_area = 923.175/(1000*1000); % m^2
stab_arm = 222/1000; % m
m = 76.6/1000; % kg (with battery in)
g = 9.81; % m/s^2
thr_to_thrust = 0.1861; % grams per unit throttle command
ail_comm_to_rad = 8.430084159809215e-04; % Multiply raw command by these to get deflection in rads
elev_comm_to_rad = -0.001473692553866; % Sign is correct
rud_comm_to_rad = -0.001846558348610; % Sign is correct
ail_area_out = (4.5718848*10^3)/(1000*1000);
ail_area_in = (1.51275*10^3)/(1000*1000);
thr_min = 270; % If it's less than 270, prop doesn't spin
rho = 1.1839; % kg/m^3 (density of air)
throttle_trim = 250; % At 250, we have 0 propwash speed (according to fit from anemometer readings)
ail_trim = 512;
rud_trim = 512;
elev_trim = 512;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Get states
rpy = x(4:6);
U = x(7);
V = x(8);
W = x(9);
P = x(10);
Q = x(11);
R = x(12);
R_body_to_world = rpy2rotmat(rpy);
R_world_to_body = R_body_to_world';
% COM velocity in world coordinates
xdots_world = R_body_to_world*[U;V;W];
% Angular velocity in world coordinate frame
omega_world = R_body_to_world*[P;Q;R];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Get control inputs
%Throttle signal is 150-850
thr = u(1) - throttle_trim; % Shift it
if thr < (thr_min - throttle_trim)
thr = 0;
end
% thr = thr_fac*max(thr,0); % Scale it and don't let it go negative
% positive AilL is negative lift (front of aileron tips downwards)
ail = (u(2)-ail_trim)*ail_comm_to_rad; % input in radians of deflection
ailL = ail; % This is correct sign
% Positive ailR is positive lift (front of aileron tips downwards)
ailR = ail; % This is correct sign
%positive elevator is deflection up (negative lift - makes plane
%pitch up)
elev = (u(3)-elev_trim)*elev_comm_to_rad; % input in radians of deflection
%positive rudder is deflection to the right
rud = (u(4)-rud_trim)*rud_comm_to_rad;% input in radians of deflection
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Now, the hard stuff
% Do translataional stuff first
% Speed of plane
% vel = sqrt(U^2 + V^2 + W^2);
velXZ = sqrt(U^2 + W^2);
velXY = sqrt(U^2 + V^2);
% Angle of attack
alpha = atan2(W,U);
%%Forces due to wings and ailerons%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Propwash over ailerons (0.2932 was fit from anemometer experiments)
upa = sqrt((velXZ^2)/4 + thr_to_sp_ail*0.2932*thr) - velXZ/2; %
uwa = thr_vel_fac_ail*sqrt(velXZ^2 + upa^2 + 2*velXZ*upa*cos(alpha));
alpha_wa = atan2(W,U + upa);
%Lift force generated by wing components. (flat plate)
%lift = dynamic pressure * area * Coefficient of lift.
left_wing_out_lift = pressure(velXZ) * wing_area_out * Cl_fp_fit(alpha);
left_wing_in_lift = pressure(uwa) * wing_area_in * Cl_fp_fit(alpha_wa);
right_wing_out_lift = pressure(velXZ) * wing_area_out * Cl_fp_fit(alpha);
right_wing_in_lift = pressure(uwa) * wing_area_in * Cl_fp_fit(alpha_wa);
%Lift force generated by ailerons. (flat plate)
%lift = dynamic pressure * area * Coefficient of lift.
left_ail_out_lift = pressure(velXZ) * ail_area_out * Cl_fp(alpha-ailL);
left_ail_in_lift = pressure(uwa) * ail_area_in * Cl_fp(alpha_wa-ailL);
right_ail_out_lift = pressure(velXZ) * ail_area_out * Cl_fp(alpha+ailR);
right_ail_in_lift = pressure(uwa) * ail_area_in * Cl_fp(alpha_wa+ailR);
%Drag force generated by wing components.
left_wing_out_drag = pressure(velXZ) * wing_area_out * Cd_fp_fit(alpha);
left_wing_in_drag = pressure(uwa) * wing_area_in * Cd_fp_fit(alpha_wa);
right_wing_out_drag = pressure(velXZ) * wing_area_out * Cd_fp_fit(alpha);
right_wing_in_drag = pressure(uwa) * wing_area_in * Cd_fp_fit(alpha_wa);
%Drag force generated by ailerons.
left_ail_out_drag = pressure(velXZ) * ail_area_out * Cd_fp(alpha-ailL);
left_ail_in_drag = pressure(uwa) * ail_area_in * Cd_fp(alpha_wa-ailL);
right_ail_out_drag = pressure(velXZ) * ail_area_out * Cd_fp(alpha+ailR);
right_ail_in_drag = pressure(uwa) * ail_area_in * Cd_fp(alpha_wa+ailR);
% Collect these forces and represent them in correct frame
F_left_wing_out = rotAlpha([left_wing_out_drag;0;left_wing_out_lift],alpha);
F_left_wing_in = rotAlpha([left_wing_in_drag;0;left_wing_in_lift],alpha_wa);
F_left_ail_out = rotAlpha([left_ail_out_drag;0;left_ail_out_lift],alpha);
F_left_ail_in = rotAlpha([left_ail_in_drag;0;left_ail_in_lift],alpha_wa);
F_right_wing_out = rotAlpha([right_wing_out_drag;0;right_wing_out_lift],alpha);
F_right_wing_in = rotAlpha([right_wing_in_drag;0;right_wing_in_lift],alpha_wa);
F_right_ail_out = rotAlpha([right_ail_out_drag;0;right_ail_out_lift],alpha);
F_right_ail_in = rotAlpha([right_ail_in_drag;0;right_ail_in_lift],alpha_wa);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Forces due to elevator
% Propwash over rudder/elevator (0.1444 was fit from anemometer
% experiments)
upe = sqrt((velXZ^2)/4 + thr_to_sp_elev*0.1444*thr) - velXZ/2;
% Elevator position and velocity
% Velocity of elevator in world coordinate frame
xdot_elev = xdots_world + cross(omega_world,R_body_to_world*[-elev_arm;0;0]);
xdot_elev_body = R_world_to_body*xdot_elev;
alpha_elev = atan2(xdot_elev_body(3),xdot_elev_body(1));
uwe = thr_vel_fac_elev*sqrt(velXZ^2 + upe^2 + 2*velXZ*upe*cos(alpha_elev));
alpha_we = atan2(xdot_elev_body(3),xdot_elev_body(1) + upe);
%include lift terms from flat plate theory of elevator
elev_lift = elev_lift_fac*pressure(uwe) * elev_area * ... likely a small term
Cl_fp(alpha_we-elev); %angle of deflection of Elevator
% NOTE: Do we need separate inner/outer terms for elevator too? (I decided
% not to do this - I think that's reasonable)
%Compute drag on elevator using flat plate theory
elev_drag = pressure(uwa) * elev_area * Cd_fp(alpha_we-elev);
% Rotate to correct frame
F_elev = rotAlpha([elev_drag;0;elev_lift],alpha_we); % elevator
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Rudder and stabilizer
%Stabilizing force from aircraft yawing (this is a crude approximation)
F_stabilizer = [0;-stab_force_fac*sign(R) * pressure(R*stab_arm) * (stab_area+rudder_area);0];
% Rudder position and velocity
% Velocity of rudder in world coordinate frame
xdot_rudder = xdots_world + cross(omega_world,R_body_to_world*[-rudder_arm;0;-9/1000]);
xdot_rudder_body = R_world_to_body*xdot_rudder;
velXY_rud = sqrt(xdot_rudder_body(1)^2 + xdot_rudder_body(2)^2);
V_rud = xdot_rudder_body(2);
% Sideslip angle for rudder
beta_rud = atan2(V,sqrt(velXY_rud^2-V_rud^2) + upe); % Assuming propwash over rudder is same as elevator
% Rudder force flat plate
rudder_force_l = pressure(uwe) * rudder_area * Cl_fp(beta_rud + rud);
rudder_force_d = pressure(uwe) * rudder_area * Cd_fp(beta_rud + rud);
rudder_force = [rudder_force_d;rudder_force_l;0];
% Rotate to correct frame
F_rudder = [-cos(beta_rud), sin(beta_rud), 0; -sin(beta_rud), -cos(beta_rud), 0;0 0 0]*rudder_force;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Gravity, thrust, and body drag
F_gravity = R_world_to_body*[0;0;m*g];
F_thrust = [thr_fac*thr_to_thrust*thr*g/1000;0;0]; % thr_to_thrust was estimated from digital scale experiments
% Drag due to body (in x-direction)
body_drag_x = body_x_drag_fac*pressure(U);
% Drag due to body in y-direction
body_drag_y = body_y_drag_fac * pressure(V) * wing_total_area; % Just a rough initial estimate
% Drag due to wing in z-direction
body_drag_z = body_z_drag_fac * pressure(W) * wing_total_area;
F_body_drag = [body_drag_x;-sign(V)*body_drag_y;-sign(W)*body_drag_z];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Additional angular rate dependent forces
F_rate_dependent = [F_Q_fac_x*Q;0;F_Q_fac_z*Q]; % These should be small effects hopefully
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Now moments/rotational stuff
% Moment from wings and ailerons
d_left_wing_out = [0;-out_dist;0];
d_left_wing_in = [0;-in_dist;0];
d_left_ail_out = [-ail_out_dist_x;-out_dist;0];
d_left_ail_in = [-ail_in_dist_x;-in_dist;0];
d_right_wing_out = [0;out_dist;0];
d_right_wing_in = [0;in_dist;0];
d_right_ail_out = [-ail_out_dist_x;out_dist;0];
d_right_ail_in = [-ail_in_dist_x;in_dist;0];
M_left_wing_out = cross(d_left_wing_out,F_left_wing_out);
M_left_wing_in = cross(d_left_wing_in,F_left_wing_in);
M_left_ail_out = cross(d_left_ail_out,F_left_ail_out);
M_left_ail_in = cross(d_left_ail_in,F_left_ail_in);
M_right_wing_out = cross(d_right_wing_out,F_right_wing_out);
M_right_wing_in = cross(d_right_wing_in,F_right_wing_in);
M_right_ail_out = cross(d_right_ail_out,F_right_ail_out);
M_right_ail_in = cross(d_right_ail_in,F_right_ail_in);
% Moment from elevator
d_elev = [-elev_arm;0;0];
M_elev = cross(d_elev,F_elev);
% Moment from stabilizer
d_stabilizer = [-stab_arm;0;-35/1000]; % -35 mm in z approximately
M_stabilizer = cross(d_stabilizer,F_stabilizer);
% Moment from rudder
d_rudder = [-rudder_arm;0;-9/1000]; % -9 mm in z approximately
M_rudder = cross(d_rudder,F_rudder);
% Moment from throttle aerodynamic drag
M_throttle = [thr_drag_fac*thr;0;0];
% Additional rate dependent moments
M_rate_dependent = [sign(P)*M_P_fac*P^2;sign(Q)*M_Q_fac*Q^2;sign(R)*M_R_fac*R^2];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Put equations together
% Kinematics
xyzdot = R_body_to_world*[U;V;W];
Phi = angularvel2rpydotMatrix(rpy);
rpydot = Phi*omega_world;
% Dynamics
% Translational equations
F_total = F_left_wing_out + F_left_wing_in + F_left_ail_out + F_left_ail_in + ... % left wing
F_right_wing_out + F_right_wing_in + F_right_ail_out + F_right_ail_in + ... % right wing
F_elev + F_stabilizer + F_rudder + ... % elevator, stabilizer, rudder
F_gravity + F_thrust + F_body_drag + F_rate_dependent; % gravity, thrust, body drag, rate dependent force
Sw = [0 -R Q; R 0 -P;-Q P 0];
UVW_dot = -Sw*[U;V;W] + F_total/m;
% Rotational stuff
M_total = M_left_wing_out + M_left_wing_in + M_left_ail_out + M_left_ail_in + ...
M_right_wing_out + M_right_wing_in + M_right_ail_out + M_right_ail_in + ...
M_elev + M_stabilizer + M_rudder + M_throttle + M_rate_dependent;
PQR_dot = invJ*(M_total - cross([P;Q;R],J*[P;Q;R]));
xdot = [xyzdot;rpydot;UVW_dot;PQR_dot];
%% Helper functions
function cl = Cl_fp(a) % Flat plate model
if a > pi/2
a = pi/2;
elseif a<-pi/2
a = -pi/2;
end
cl = 2*sin(a)*cos(a);
end
function cd = Cd_fp(a) % Flat plate
if a > pi/2
a = pi/2;
elseif a<-pi/2
a = -pi/2;
end
cd = 2*(sin(a)^2);
end
function cl = Cl_fp_fit(a) % Flat plate model with correction terms
if a > pi/2
a = pi/2;
elseif a<-pi/2
a = -pi/2;
end
% These numbers were fit from no-throttle experiments
cl = 2*sin(a)*cos(a) + 0.6602*sin(2.8717*a);
end
function cd = Cd_fp_fit(a) % Flat plate with correction terms
if a > pi/2
a = pi/2;
elseif a<-pi/2
a = -pi/2;
end
% These numbers were fit from no-throttle experiments
cd = 2*(sin(a)^2) + 0.8296*(sin(a)^2) + 0.1372;
end
function f = rotAlpha(f,a)
% Rotation matrix
Ra = [-cos(a) 0 sin(a); ...
0 0 0 ; ...
-sin(a) 0 -cos(a)];
% Rotate f using Ra
f = Ra*f;
end
function pre = pressure(vel) %Dynamic Pressure = .5 rho * v^2
pre = .5 * 1.1839 * vel^2; % N/m^2
end
end
function y = output(obj,t,x,u)
y = x;
end
function x = getInitialState(obj)
x = zeros(12,1);
end
end
end